On the Bloch-Kato conjecture for Hecke \(L\)-functions
type: Item
instance of: scholarly article
title: On the Bloch-Kato conjecture for Hecke \(L\)-functions
zbMATH Open document ID: 0869.11055
author: Li Guo
published in: Journal of Number Theory
publication date: 1996-06-04T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the ring of integers of an imaginary quadratic field \(K\). Let \(\psi\) be the Größencharacter attached to \(E/K\); for \(k>j>0\), \(\psi^k \overline \psi^{-j}\) is a Größencharacter. The author studies the \(p\)-part of the conjecture of Bloch-Kato for the special value of the Hecke \(L\)-function \(L(\psi^{-k} \overline \psi^j,s)\) at \(s=0\) when the prime \(p\) splits in \(K\) and \(p>k+1\). The Block-Kato conjecture may be considered as a vast generalization, for motives, of the conjecture of Birch and Swinnerton-Dyer on elliptic curves. In the present case, the motive \({\mathcal M}= {\mathcal M}_{k,j}\) comes from the Größencharacter \(\psi^k \overline \psi^{-j}\) and its motivic \(L\)-function coincides with the Hecke \(L\)-function \(L(\psi^{-k} \overline \psi^j, s)\). If \(L(\psi^{-k} \overline \psi^j,0)\neq 0\), the author shows that the \(p\)-part of the Bloch-Kato conjecture giving the Tamagawa number of the motive \({\mathcal M}\) is true, and is in fact equivalent to a formula for the special values of \(L(\psi^{-k} \overline \psi^j,s)\) previously proved by him [Math. Ann. 297, 221-233 (1993; Zbl 0789.14018)].
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 874201
zbMATH Keywords: elliptic curve with complex multiplication
zbMATH Keywords: special value of the Hecke \(L\)-function
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: motivic \(L\)-function
zbMATH Keywords: \(p\)-part of the Bloch-Kato conjecture
zbMATH Keywords: motives
zbMATH Keywords: Größencharacter
zbMATH Keywords: Tamagawa number
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: jnth.1996.0053
OpenAlex ID: W1965303726
Wikidata QID: Q122956377
DOI: 10.1006/JNTH.1996.0053
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description: scientific article; zbMATH DE number 874201